Ok so this was a pretty fun one for me. Since both parties have been in existence, they have held their conventions in the same city a total of five times.

Year	City	Days apart	Same Complex?
1884	Chicago	32	Maybe?
1944	Chicago	24	YES!
1948	Philadelphia	17	YES!
1952	Chicago	10	YES!
1972	Miami Beach	39	YES!

So the closest that that the conventions have ever been was in 1952 when they were held at the International Amphitheatre in Chicago just ten days apart. The '52 conventions were also significant in that they were the first ones to be nationally broadcasted.

I hope this helps!